Metrics in the sphere of a C*-module

Esteban Andruchow, and Alejandro Varela. "Metrics in the sphere of a C*-module." Open Mathematics 5.4 (2007): 639-653. <http://eudml.org/doc/269531>.

@article{EstebanAndruchow2007,
abstract = {Given a unital C*-algebra \[\mathcal \{A\}\] and a right C*-module \[\mathcal \{X\}\] over \[\mathcal \{A\}\] , we consider the problem of finding short smooth curves in the sphere \[\mathcal \{S\}\_\mathcal \{X\} \] = x ∈ \[\mathcal \{X\}\] : 〈x, x〉 = 1. Curves in \[\mathcal \{S\}\_\mathcal \{X\} \] are measured considering the Finsler metric which consists of the norm of \[\mathcal \{X\}\] at each tangent space of \[\mathcal \{S\}\_\mathcal \{X\} \] . The initial value problem is solved, for the case when \[\mathcal \{A\}\] is a von Neumann algebra and \[\mathcal \{X\}\] is selfdual: for any element x 0 ∈ \[\mathcal \{S\}\_\mathcal \{X\} \] and any tangent vector ν at x 0, there exists a curve γ(t) = e tZ(x 0), Z ∈ \[\mathcal \{L\}\_\mathcal \{A\} (\mathcal \{X\})\] , Z* = −Z and ∥Z∥ ≤ π, such that γ(0) = x 0 and \[\dot\{\gamma \}\] (0) = ν, which is minimizing along its path for t ∈ [0, 1]. The existence of such Z is linked to the extension problem of selfadjoint operators. Such minimal curves need not be unique. Also we consider the boundary value problem: given x 0, x 1 ∈ \[\mathcal \{S\}\_\mathcal \{X\} \] , find a curve of minimal length which joins them. We give several partial answers to this question. For instance, let us denote by f 0 the selfadjoint projection I − x 0 ⊗ x 0, if the algebra f 0 \[\mathcal \{L\}\_\mathcal \{A\} (\mathcal \{X\})\] f 0 is finite dimensional, then there exists a curve γ joining x 0 and x 1, which is minimizing along its path.},
author = {Esteban Andruchow, Alejandro Varela},
journal = {Open Mathematics},
keywords = {C*-modules; spheres; geodesics; -modules},
language = {eng},
number = {4},
pages = {639-653},
title = {Metrics in the sphere of a C*-module},
url = {http://eudml.org/doc/269531},
volume = {5},
year = {2007},
}

TY - JOUR
AU - Esteban Andruchow
AU - Alejandro Varela
TI - Metrics in the sphere of a C*-module
JO - Open Mathematics
PY - 2007
VL - 5
IS - 4
SP - 639
EP - 653
AB - Given a unital C*-algebra \[\mathcal {A}\] and a right C*-module \[\mathcal {X}\] over \[\mathcal {A}\] , we consider the problem of finding short smooth curves in the sphere \[\mathcal {S}_\mathcal {X} \] = x ∈ \[\mathcal {X}\] : 〈x, x〉 = 1. Curves in \[\mathcal {S}_\mathcal {X} \] are measured considering the Finsler metric which consists of the norm of \[\mathcal {X}\] at each tangent space of \[\mathcal {S}_\mathcal {X} \] . The initial value problem is solved, for the case when \[\mathcal {A}\] is a von Neumann algebra and \[\mathcal {X}\] is selfdual: for any element x 0 ∈ \[\mathcal {S}_\mathcal {X} \] and any tangent vector ν at x 0, there exists a curve γ(t) = e tZ(x 0), Z ∈ \[\mathcal {L}_\mathcal {A} (\mathcal {X})\] , Z* = −Z and ∥Z∥ ≤ π, such that γ(0) = x 0 and \[\dot{\gamma }\] (0) = ν, which is minimizing along its path for t ∈ [0, 1]. The existence of such Z is linked to the extension problem of selfadjoint operators. Such minimal curves need not be unique. Also we consider the boundary value problem: given x 0, x 1 ∈ \[\mathcal {S}_\mathcal {X} \] , find a curve of minimal length which joins them. We give several partial answers to this question. For instance, let us denote by f 0 the selfadjoint projection I − x 0 ⊗ x 0, if the algebra f 0 \[\mathcal {L}_\mathcal {A} (\mathcal {X})\] f 0 is finite dimensional, then there exists a curve γ joining x 0 and x 1, which is minimizing along its path.
LA - eng
KW - C*-modules; spheres; geodesics; -modules
UR - http://eudml.org/doc/269531
ER -